If you are considering human vision there is a definite (and surprisingly small) number of distinguishable colors.

This is known as a MacAdam diagram and shows a region around a single colour, on a chromaticity diagram, that is indistinguishable from the color at the centre.     
The total number of colors would be the number of ellipses needed to completely fill the coor space.  Obviously this depends on the individual, lighting, etc
 
![enter image description here][1]


  [1]: https://i.sstatic.net/LtlAA.png